Tight closure

Remarks

The ideal $I^{[q]}$ is the ideal generated by the image of $I$ under the $e$-th power of the Frobenius endomorphism. If $I = (a_1, \ldots, a_m)$, then $I^{[q]} = (a_1^q, \ldots, a_m^q)$. $I^*$ is an ideal, $I \subseteq I^*$, and $(I^*)^* = I^*$. The ideal $I$ is tightly closed if $I^* = I$.

Example: let $k$ be a field of characteristic $p$, let $R = k[x, y]/(x^2)$, and let $I = (0)$. The only minimal prime is $(x)$, so $1 \in R^\circ$, and $1 \cdot x^q = 0 \in I^{[q]}$ for $q \ge 2$. So $x \in (0)^*$; more generally every nilpotent element lies in $(0)^*$.

Non-example: let $R = k[x]$ and $I = (x^2)$; then $x \notin I^*$. Indeed $c x^q \in (x^{2q})$ means $x^q$ divides $c$, and if this holds for infinitely many $q$ then $c = 0$, which is not in $R^\circ$.